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Cool Math in Real Life: The Mathematics of Buying a House

Posted by Chris Brida

Jan 6, 2016 7:00:00 AM

In my last blog, I discussed three fundamental concepts in Shark Tank—valuation, royalties and profit margins. These concepts are super important for making your watching experience a little bit more wholesome (like a good chicken noodle soup…or something).


Armed with those three tools, you can understand a bit more about what’s going on in the show—how much the companies think they are worth, how much the sharks are offering to take from them, and how much money the company is currently bringing in. The world of business is chock full of real-life math. You don’t need an MBA (Masters in Business Administration) to dive right into the world of numbers and money. Just read this blog (disclaimer: if you start a business based on my blog, I will take all of the credit for your success, and none of the credit for your loss)!

Since buying a house is one of the biggest financial decisions a person might make, today I’m going to go through how to calculate your monthly mortgage payments.

Calculating Mortgage


Listen up homeowners, or students who want to buy a house in the future! Do you know what’s actually happening with your mortgage payments? There’s some really simple math behind the actual calculation that a mortgage lender makes to determine your monthly payment.

Even if you aren’t a homeowner and you are a student, this formula can still be applied to buying a car or anything that has an interest payment. Unfortunately, in the world of finance, nothing is ever as simple as: I take out a $100 loan and I make ten $10 payments back to the loan provider and badah bing, done. No, no, no—this just isn’t how it works.

Instead, mortgages and other loans are calculated using compounding interest, which actually increase the amount of money you have to pay over time. It’s almost like borrowing $20 from a friend and the classic example of “I’ll pay you back with interest” so you give him $21 back. Almost. Let’s dive right in.

The standard formula for calculating monthly mortgage payments is:

M = P [ i(1 + i)n ] / [ (1 + i)n - 1]

Where:

→ M is the monthly payment

→ P is the principle (or the total amount originally loaned) of the loan

→ i is the interest rate per month

→ n is the number of number of monthly payments to be made over the life of the loan 

Let’s assume an example of a $100,000 mortgage that is compounded monthly (this means they will charge interest on the loan every month) at a 5% rate for 15 years.

First we would solve for i by taking our interest rate as a decimal (0.05) and dividing it by 12 (because we are making monthly payments and there are 12 months in the year) 

0.05 / 12 = .004167

This means that every month, we will be charged about .4167% interest, so that the yearly total is 5% (.4167% x 12 ~ 5%). 

We calculate n by taking the number of payments we have to make in one year (12) and multiplying by the number of years we anticipate paying for (15)

12 x 15 = 180

This means that we will make 180 equal payments on our mortgage throughout the next year.

 

 

Next we are going to solve for (1+i)n which is the calculation for finding interest payments for the next 180 payments. This is the compounding portion of our formula—it means that we will be charged .004167% 180 times.

With that we have (1+.004167)180 which comes out to be 2.11383

Now our formula reads: M = P [(.004167)(2.11383)] / [2.11383 – 1]

Remember we substituted for i and n in our initial formula, and did a bit of simplifying...

We can now add in the loan amount ($100,000) to determine our monthly cost:

M = 100000[(.004167x2.11383) / 1.11383

Some quick math andddddd—

We calculate a monthly payment of $790.81

 

Now what can we do with that? Well, we can take that number ($790.81) and multiply it by the number of payments that we will make to determine how much we will pay in interest over the life of the loan. In this case, 

$790.81 x 180 = $142345.80

Since our initial loan was $100,000, we are paying $42345.80 in interest over the 15 years. So much for owing that friend just one dollar!

 

 

Let’s try another example just to make sure we’ve got it: Let’s say you have a $200,000 loan with a 4% interest rate compounded monthly for 30 years!

The interest calculation would be .04 / 12 = .00333 

The total payments (n) would be 12 x 30 = 360

So our formula will be written as M = 200000[.00333 x (1+.00333)360] / [(1+.00333)360 – 1]

Simplification gets us: 200000 [.00333 x 3.3095] / [2.3095]

Finishing up the math, we have monthly payments of $954.37

Over the life of the loan, this comes out to:

$957.37 x 360 = 343574.68 - $200000

We will pay nearly $143000 in interest on this house!

Yikes!


 

If you want to see if you can afford a house in the not-so-distant future, use this formula to figure out if you can afford the monthly payments or not. I wish I could tell you it is as easy as just paying for the loan, but banks need to make money too! 

Mathematics is behind all of the financial decisions we make, such as which credit cards to use due to the interest we’ll be charged, or whether to buy a new or used car (did you know that the moment you drive a car off the lot, it’s not worth nearly as much as you just paid?), and who to buy it from. Some people get intimidated by mathematics in real life, but with a little work—and a few blog posts like this!—you can make sure you stay ahead of the game. 

Tune in next time for a discussion on amortization, which is the repayment of a loan’s principle over time!

Tip for Teachers:

Use the examples from this blog as real world applications of math in your next math class, to help your students understand the ways in which mathematics is interwoven with our daily lives—and can help them succeed at business!


About the Author

Chris_Brida_Headshot-1Chris Brida is a mathematics teacher in Baltimore, MD. He currently teaches 9th grade Algebra. Chris is a regular guest contributor to our blog, and we feel lucky to have him. 

 

A note on guest posts: Our community blog is a place for educators from all walks of life to share opinions and exchange ideas. Simply because a post appears here does not necessarily mean we endorse the views presented therein. That being said, we'd love to hear what you think! Please post any questions or comments below, and we'll get right back. 

Topics: Teaching & Learning, STEM, Innovation, Inspiration, Educational Technologies

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